Related papers: H\"older continuity and differentiability on subse…

A diffeomorphism f is called super exponential divergent if for every r>1, the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity, where Per_n(f) is the set of all periodic points of f with period n. This property is…

Let $N\in\mathbb{N}$ and $u$ be a weak solution of equation $\displaystyle Lu\equiv - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{j}}(\frac{\partial u}{\partial x_{i}}b^{ij})= f$ in $\Omega\subset \mathbb{R}^{N}$. We obtain functions $G$…

We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, \] where $\phi$ satisfies a…

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize…

A multiplicative $\alpha$-spanner $H$ is a subgraph of $G=(V,E)$ with the same vertices and fewer edges that preserves distances up to the factor $\alpha$, i.e., $d_H(u,v)\leq\alpha\cdot d_G(u,v)$ for all vertices $u$, $v$. While many…

The gradient of any local minimiser of functionals of the type $$ w \mapsto \int_\Omega f(x,w,Dw)\,dx+\int_\Omega w\mu\,dx, $$ where $f$ has $p$-growth, $p>1$, and $\Omega \subset \mathbb R^n$, is continuous provided the optimal Lorentz…

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…

After introducing the concept of functional dissipativity of the Dirichlet problem in a domain $\Omega\subset {\mathbb R}^N$ for systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$…

Let $0<\alpha<d$ and $1\leq p<d/\alpha$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_\alpha f$ are weakly differentiable and $…

We develop the notion of higher Cheeger constants for a measurable set $\Omega \subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(\Omega) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken…

We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If…

We consider the class of diffeomorphisms of a manifold that its differential keeps invariant a one-dimensional subbundle $E$. For that type of diffeomorphisms is naturally defined a one-parameter family called $E-$translation. We prove that…

In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any…

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be H\"older continuous. The…

We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the…

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…

Local H\"older regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling…

For a given domain $\Omega \subset \Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower…

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic…

Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2\pi i nt}/n$, for a large…